advances in generalized blockmodeling pat doreianvlado.fmf.uni-lj.si/pub...

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Advances in Generalized BlockmodelingPat Doreian

• V. Batagelj, P. Doreian, A. Ferligoj, A. Mrvar:Generalized Blockmodeling

• P. Doreian, Hyung Sam Park:Networks of Environmental Social Movement Organizations: TheTurning Point Project

• A. Ziberna:Generalized Blockmodeling for Valued Networks

• N. Kejzar, V. Batagelj:Analysis of US Patents Network: Development of Patents overTime

version: February 18, 2005 / 20 : 24

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GeneralizedBlockmodelingVladimir BatageljPatrick DoreianAnuska FerligojAndrej Mrvar

SUNBELT XXVRedondo Beach, CA, February 18, 2005

version: February 18, 2005 / 20 : 24

V. Batagelj, P. Doreian, A. Ferligoj, A. Mrvar: Advances in Generalized Blockmodeling 3'

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Outline1 Advances in Generalized Blockmodeling

Pat Doreian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Matrix rearrangement view on blockmodeling . . . . . . . . . . . . . . . . . . . 1

3 Blockmodeling as a clustering problem . . . . . . . . . . . . . . . . . . . . . . 3

21 Pre-specified blockmodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

30 Signed graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

36 Blockmodeling in 2-mode networks . . . . . . . . . . . . . . . . . . . . . . . . 36

39 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

SUNBELT XXV, Redondo Beach, CA, February 18, 2005 s s y s l s y ss * 6


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Matrix rearrangement view on blockmodelingSnyder & Kick’s World trade network / n = 118, m = 514

Pajek - shadow 0.00,1.00 Sep- 5-1998World trade - alphabetic order

afg alb alg arg aus aut bel bol bra brm bul bur cam can car cha chd chi col con cos cub cyp cze dah den dom ecu ege egy els eth fin fra gab gha gre gua gui hai hon hun ice ind ins ire irn irq isr ita ivo jam jap jor ken kmr kod kor kuw lao leb lib liy lux maa mat mex mla mli mon mor nau nep net nic nig nir nor nze pak pan par per phi pol por rum rwa saf sau sen sie som spa sri sud swe swi syr tai tha tog tri tun tur uga uki upv uru usa usr ven vnd vnr wge yem yug zai

afg

a

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alg

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bul

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Pajek - shadow 0.00,1.00 Sep- 5-1998World Trade (Snyder and Kick, 1979) - cores

uki net bel lux fra ita den jap usa can bra arg ire swi spa por wge ege pol aus hun cze yug gre bul rum usr fin swe nor irn tur irq egy leb cha ind pak aut cub mex uru nig ken saf mor sud syr isr sau kuw sri tha mla gua hon els nic cos pan col ven ecu per chi tai kor vnr phi ins nze mli sen nir ivo upv gha cam gab maa alg hai dom jam tri bol par mat alb cyp ice dah nau gui lib sie tog car chd con zai uga bur rwa som eth tun liy jor yem afg mon kod brm nep kmr lao vnd

uki

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Alphabetic order of countries (left) and rearrangement (right)



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Ordering the matrix

There are several ways how to rearrange a given matrix – determine anordering or permutation of its rows and columns – to get some insight intoits structure:

• ordering by degree;

• ordering by connected components;

• ordering by core number, connected components inside core levels, anddegree;

• ordering according to a hierarchical clustering and some other property.

There exists also some special procedures to determine the ordering such asseriation and clumping (Murtagh).



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Blockmodeling as a clustering problem

The goal of blockmodeling is to reducea large, potentially incoherent networkto a smaller comprehensible structurethat can be interpreted more readily.Blockmodeling, as an empirical proce-dure, is based on the idea that units ina network can be grouped according tothe extent to which they are equivalent,according to some meaningful defini-tion of equivalence.



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Cluster, clustering, blocks

One of the main procedural goals of blockmodeling is to identify, in a givennetwork N = (U, R), R ⊆ U ×U, clusters (classes) of units that sharestructural characteristics defined in terms of R. The units within a clusterhave the same or similar connection patterns to other units. They form aclustering C = {C1, C2, . . . , Ck} which is a partition of the set U. Eachpartition determines an equivalence relation (and vice versa). Let us denoteby ∼ the relation determined by partition C.

A clustering C partitions also the relation R into blocks

R(Ci, Cj) = R ∩ Ci × Cj

Each such block consists of units belonging to clusters Ci and Cj and allarcs leading from cluster Ci to cluster Cj . If i = j, a block R(Ci, Ci) iscalled a diagonal block.



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Structural and regular equivalence

Regardless of the definition of equivalence used, there are two basicapproaches to the equivalence of units in a given network (compare Faust,1988):

• the equivalent units have the same connection pattern to the sameneighbors;

• the equivalent units have the same or similar connection pattern to(possibly) different neighbors.

The first type of equivalence is formalized by the notion of structuralequivalence and the second by the notion of regular equivalence with thelatter a generalization of the former.



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Structural equivalence

Units are equivalent if they are connected to the rest of the network inidentical ways (Lorrain and White, 1971). Such units are said to bestructurally equivalent.

The units X and Y are structurally equivalent, we write X ≡ Y, iff thepermutation (transposition) π = (X Y) is an automorphism of the relationR (Borgatti and Everett, 1992).

In other words, X and Y are structurally equivalent iff:

s1. XRY ⇔ YRX s3. ∀Z ∈ U \ {X,Y} : (XRZ ⇔ YRZ)s2. XRX ⇔ YRY s4. ∀Z ∈ U \ {X,Y} : (ZRX ⇔ ZRY)



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. . . structural equivalence

The blocks for structural equivalence are null or complete with variationson diagonal in diagonal blocks.



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Regular equivalence

Integral to all attempts to generalize structural equivalence is the idea thatunits are equivalent if they link in equivalent ways to other units that arealso equivalent.

White and Reitz (1983): The equivalence relation ≈ on U is a regularequivalence on network N = (U, R) if and only if for all X,Y,Z ∈ U,X ≈ Y implies both

R1. XRZ ⇒ ∃W ∈ U : (YRW ∧W ≈ Z)R2. ZRX ⇒ ∃W ∈ U : (WRY ∧W ≈ Z)

Another view of regular equivalence is based on colorings (Everett, Borgatti1996).



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. . . regular equivalence

Theorem 1 (Batagelj, Doreian, Ferligoj, 1992) Let C = {Ci} be apartition corresponding to a regular equivalence ≈ on the network N =(U, R). Then each block R(Cu, Cv) is either null or it has the propertythat there is at least one 1 in each of its rows and in each of its columns.Conversely, if for a given clustering C, each block has this property thenthe corresponding equivalence relation is a regular equivalence.

The blocks for regular equivalence are null or 1-covered blocks.



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Establishing Blockmodels

The problem of establishing a partition of units in a network in terms of aselected type of equivalence is a special case of clustering problem that canbe formulated as an optimization problem (Φ, P ) as follows:

Determine the clustering C? ∈ Φ for which

P (C?) = minC∈Φ

P (C)

where Φ is the set of feasible clusterings and P is a criterion function.

Since the set of units U is finite, the set of feasible clusterings is alsofinite. Therefore the set Min(Φ, P ) of all solutions of the problem (optimalclusterings) is not empty.



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Criterion function

Criterion functions can be constructed

• indirectly as a function of a compatible (dis)similarity measure betweenpairs of units, or

• directly as a function measuring the fit of a clustering to an idealone with perfect relations within each cluster and between clustersaccording to the considered types of connections (equivalence).

Criterion function P (C) has to be sensitive to considered equivalence:

P (C) = 0 ⇔ C defines considered equivalence.



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Generalized BlockmodelingA blockmodel consists of structures ob-tained by identifying all units from thesame cluster of the clustering C. Foran exact definition of a blockmodel wehave to be precise also about whichblocks produce an arc in the reducedgraph and which do not, and of whattype. Some types of connections arepresented in the figure on the next slide.The reduced graph can be representedby relational matrix, called also imagematrix.



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Block Types



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Generalized equivalence / Block Types

Y1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1complete

Y0 1 0 0 0

X 1 1 1 1 10 0 0 0 00 0 0 1 0

row-dominant

Y0 0 1 0 0

X 0 0 1 1 01 1 1 0 00 0 1 0 1

col-dominant

Y0 1 0 0 0

X 1 0 1 1 00 0 1 0 11 1 0 0 0

regular

Y0 1 0 0 0

X 0 1 1 0 01 0 1 0 00 1 0 0 1

row-regular

Y0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0col-regular

Y0 0 0 0 0

X 0 0 0 0 00 0 0 0 00 0 0 0 0

null

Y0 0 0 1 0

X 0 0 1 0 01 0 0 0 00 0 0 1 0

row-functional

Y1 0 0 00 1 0 0

X 0 0 1 00 0 0 00 0 0 1

col-functional



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Characterizations of Types of Blocksnull nul all 0 ∗

complete com all 1 ∗

regular reg 1-covered rows and columns

row-regular rre each row is 1-covered

col-regular cre each column is 1 -covered

row-dominant rdo ∃ all 1 row ∗

col-dominant cdo ∃ all 1 column ∗

row-functional rfn ∃! one 1 in each row

col-functional cfn ∃! one 1 in each column

non-null one ∃ at least one 1∗ except this may be diagonal

A block is symmetric iff ∀X,Y ∈ Ci × Cj : (XRY ⇔ YRX).



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Block Types and Matrices

1 1 1 1 1 1 0 0

1 1 1 1 0 1 0 1

1 1 1 1 0 0 1 0

1 1 1 1 1 0 0 0

0 0 0 0 0 1 1 1

0 0 0 0 1 0 1 1

0 0 0 0 1 1 0 1

0 0 0 0 1 1 1 0

C1 C2

C1 complete regular

C2 null complete



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Criterion function

One of the possible ways of constructing a criterion function that directlyreflects the considered equivalence is to measure the fit of a clustering toan ideal one with perfect relations within each cluster and between clustersaccording to the considered equivalence.

Given a clustering C = {C1, C2, . . . , Ck}, let B(Cu, Cv) denote the set ofall ideal blocks corresponding to block R(Cu, Cv). Then the global error ofclustering C can be expressed as

P (C) =∑

Cu,Cv∈C

minB∈B(Cu,Cv)

d(R(Cu, Cv), B)

where the term d(R(Cu, Cv), B) measures the difference (error) betweenthe block R(Cu, Cv) and the ideal block B. d is constructed on the basis ofcharacterizations of types of blocks. The function d has to be compatiblewith the selected type of equivalence.



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. . . criterion function

For example, for structural equivalence, the term d(R(Cu, Cv), B) can beexpressed, for non-diagonal blocks, as

d(R(Cu, Cv), B) =∑

X∈Cu,Y∈Cv

|rX Y − bX Y|.

where rX Y is the observed tie and bX Y is the corresponding value in anideal block. This criterion function counts the number of 1s in erstwhilenull blocks and the number of 0s in otherwise complete blocks. These twotypes of inconsistencies can be weighted differently.

Determining the block error, we also determine the type of the best fittingideal block (the types are ordered).

The criterion function P (C) is sensitive iff P (C) = 0 ⇔ µ (determinedby C) is an exact blockmodeling. For all presented block types sensitivecriterion functions can be constructed (Batagelj, 1997).



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Solving the blockmodeling problem

The obtained optimization problem can be solved by local optimization.

Once a partitioning µ and types of connection π are determined, we canalso compute the values of connections by using averaging rules.



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Benefits from Optimization Approach• ordinary / inductive blockmodeling: Given a network N and set of

types of connection T , determine the model M;

• evaluation of the quality of a model, comparing different models,analyzing the evolution of a network (Sampson data, Doreian andMrvar 1996; states / continents): Given a network N, a model M, andblockmodeling µ, compute the corresponding criterion function;

• model fitting / deductive blockmodeling: Given a network N, set oftypes T , and a family of models, determine µ which minimizes thecriterion function (Batagelj, Ferligoj, Doreian, 1998).

• we can fit the network to a partial model and analyze the residualafterward;

• we can also introduce different constraints on the model, for example:units X and Y are of the same type; or, types of units X and Y are notconnected; . . .



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Pre-specified blockmodelingIn the previous slides the inductive approaches for establishing blockmodelsfor a set of social relations defined over a set of units were discussed.Some form of equivalence is specified and clusterings are sought that areconsistent with a specified equivalence.

Another view of blockmodeling is deductive in the sense of starting with ablockmodel that is specified in terms of substance prior to an analysis.

In this case given a network, set of types of ideal blocks, and a reducedmodel, a solution (a clustering) can be determined which minimizes thecriterion function.



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Pre-Specified Blockmodels

The pre-specified blockmodeling starts with a blockmodel specified, interms of substance, prior to an analysis. Given a network, a set of idealblocks is selected, a family of reduced models is formulated, and partitionsare established by minimizing the criterion function.

The basic types of models are:

* *

* 0

* 0

* *

* 0

0 *

0 *

* 0

center - hierarchy clustering bipartition

periphery



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Prespecified blockmodeling exampleSupport network among informatics students

The analyzed network consists of social support exchange relation amongfifteen students of the Social Science Informatics fourth year class(2002/2003) at the Faculty of Social Sciences, University of Ljubljana.Interviews were conducted in October 2002.

Support relation among students was identified by the following question:

Introduction: You have done several exams since you are in thesecond class now. Students usually borrow studying material fromtheir colleagues.

Enumerate (list) the names of your colleagues that you have mostoften borrowed studying material from. (The number of listedpersons is not limited.)



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Class network

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

class.net

Vertices represent studentsin the class; circles – girls,squares – boys. Oppositepairs of arcs are replaced byedges.



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Prespecified blockmodeling example

We expect that center-periphery model exists in the network: some studentshaving good studying material, some not.

Prespecified blockmodel: (com/complete, reg/regular, -/null block)

1 2

1 [com reg] -

2 [com reg] -

Using local optimization we get the partition:

C = {{b02, b03, b51, b85, b89, b96, g09},

{g07, g10, g12, g22, g24, g28, g42, g63}}



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2 clusters solution

b02

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b85

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ModelPajek - shadow [0.00,1.00]

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Image and Error Matrices:

1 2

1 reg -

2 reg -

1 2

1 0 3

2 0 2

Total error = 5center-periphery



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The Student Government at the University of Ljubljana in1992

The relation is determined by the following question (Hlebec, 1993):

Of the members and advisors of the Student Government, whomdo you most often talk with about the matters of the StudentGovernment?

m p m m m m m m a a a1 2 3 4 5 6 7 8 9 10 11

minister 1 1 · 1 1 · · 1 · · · · ·p.minister 2 · · · · · · · 1 · · ·minister 2 3 1 1 · 1 · 1 1 1 · · ·minister 3 4 · · · · · · 1 1 · · ·minister 4 5 · 1 · 1 · 1 1 1 · · ·minister 5 6 · 1 · 1 1 · 1 1 · · ·minister 6 7 · · · 1 · · · 1 1 · 1minister 7 8 · 1 · 1 · · 1 · · · 1adviser 1 9 · · · 1 · · 1 1 · · 1adviser 2 10 1 · 1 1 1 · · · · · ·adviser 3 11 · · · · · 1 · 1 1 · ·

minister1

pminister

minister2

minister3

minister4

minister5

minister6

minister7

advisor1

advisor2

advisor3



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A Symmetric Acyclic Blockmodel of Student Government

The obtained clustering in 4clusters is almost exact. Theonly error is produced by thearc (a3,m5).



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Signed graphsA signed graph is an ordered pair (G, σ) where

• G = (V,R) is a directed graph (without loops) with set of vertices V

and set of arcs R ⊆ V × V ;

• σ : R → {p, n} is a sign function. The arcs with the sign p are positiveand the arcs with the sign n are negative. We denote the set of allpositive arcs by R+ and the set of all negative arcs by R−.

The case when the graph is undirected can be reduced to the case of directedgraph by replacing each edge e by a pair of opposite arcs both signed withthe sign of the edge e.



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Balanced and clusterable signed graphs

The signed graphs were introduced in Harary, 1953 and later studied byseveral authors. Following Roberts (1976, p. 75–77) a signed graph (G, σ)is:

• balanced iff the set of vertices V can be partitioned into two subsetsso that every positive arc joins vertices of the same subset and everynegative arc joins vertices of different subsets.

• clusterable iff the set of V can be partitioned into subsets, calledclusters, so that every positive arc joins vertices of the same subset andevery negative arc joins vertices of different subsets.



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. . . Properties

The (semi)walk on the signed graph is positive iff it contains an evennumber of negative arcs; otherwise it is negative.

The balanced and clusterable signed graphs are characterised by thefollowing theorems:

THEOREM 1. A signed graph (G, σ) is balanced iff every closed semiwalkis positive.

THEOREM 2. A signed graph (G, σ) is clusterable iff G contains noclosed semiwalk with exactly one negative arc.



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Clusterability and blockmodeling

To the sign graph clusterability problemu corespond three types of blocks:

• null all elements in a block are 0;

• positive all elements in a block are positive or 0;

• negative all elements in a block are negative or 0;

If a graph is clusterable the blocks determined by the partition are: positiveor null on the diagonal; and negative or null outside the diagonal.

The clusterability of partition C = {C1, C2, . . . , Ck} can be thereforemeasured as follows ( 0 ≤ α ≤ 1 ):

Pα(C) = α∑C∈C

∑u,v∈C

max(0,−wuv)+(1−α)∑

C,C′∈C

C 6=C′

∑u∈C,v∈C′

max(0, wuv)

The blockmodeling problem can be solved by local optimization.



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Slovenian political parties 1994 (S. Kropivnik)

1 2 3 4 5 6 7 8 9 10SKD 1 0 -215 114 -89 -77 94 -170 176 117 -210ZLSD 2 -215 0 -217 134 77 -150 57 -253 -230 49SDSS 3 114 -217 0 -203 -80 138 -109 177 180 -174LDS 4 -89 134 -203 0 157 -142 173 -241 -254 23ZSESS 5 -77 77 -80 157 0 -188 170 -120 -160 -9ZS 6 94 -150 138 -142 -188 0 -97 140 116 -106DS 7 -170 57 -109 173 170 -97 0 -184 -191 -6SLS 8 176 -253 177 -241 -120 140 -184 0 235 -132SPS-SNS 9 117 -230 180 -254 -160 116 -191 235 0 -164SNS 10 -210 49 -174 23 -9 -106 -6 -132 -164 0

SKD – Slovene Christian Democrats; ZLSD – Associated List of Social Democrats; SDSS – Social Democratic Party of Slovenia;LDS – Liberal Democratic Party; ZSESS and ZS – two Green Parties, separated after 1992 elections; DS – Democratic Party;SLS – Slovene People’s Party; SNS – Slovene National Party; SPS SNS – a group of deputies, former members of SNS, separated after 1992 elections

Network Stranke94.


http://vlado.fmf.uni-lj.si/pub/networks/data/soc/samo/stranke94.htm


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Slovenian political parties 1994 / reordered

1 3 6 8 9 2 4 5 7 10SKD 1 0 114 94 176 117 -215 -89 -77 -170 -210SDSS 3 114 0 138 177 180 -217 -203 -80 -109 -174ZS 6 94 138 0 140 116 -150 -142 -188 -97 -106SLS 8 176 177 140 0 235 -253 -241 -120 -184 -132SPS-SNS 9 117 180 116 235 0 -230 -254 -160 -191 -164ZLSD 2 -215 -217 -150 -253 -230 0 134 77 57 49LDS 4 -89 -203 -142 -241 -254 134 0 157 173 23ZSESS 5 -77 -80 -188 -120 -160 77 157 0 170 -9DS 7 -170 -109 -97 -184 -191 57 173 170 0 -6SNS 10 -210 -174 -106 -132 -164 49 23 -9 -6 0

S. Kropivnik, A. Mrvar: An Analysis of the Slovene Parliamentary Parties Network. in Developments in data analysis, MZ 12,FDV, Ljubljana, 1996, p. 209-216.



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Blockmodeling in 2-mode networksWe already presented some ways of rearranging 2-mode network matricesat the beginning of this lecture.

It is also possible to formulate this goal as a generalized blockmodelingproblem where the solutions consist of two partitions — row-partition andcolumn-partition.



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Supreme Court Voting for Twenty-Six Important DecisionsIssue Label Br Gi So St OC Ke Re Sc ThPresidential Election PE - - - - + + + + +

Criminal Law CasesIllegal Search 1 CL1 + + + + + + - - -Illegal Search 2 CL2 + + + + + + - - -Illegal Search 3 CL3 + + + - - - - + +Seat Belts CL4 - - + - - + + + +Stay of Execution CL5 + + + + + + - - -

Federal Authority CasesFederalism FA1 - - - - + + + + +Clean Air Action FA2 + + + + + + + + +Clean Water FA3 - - - - + + + + +Cannabis for Health FA4 0 + + + + + + + +United Foods FA5 - - + + - + + + +NY Times Copyrights FA6 - + + - + + + + +

Civil Rights CasesVoting Rights CR1 + + + + + - - - -Title VI Disabilities CR2 - - - - + + + + +PGA v. Handicapped Player CR3 + + + + + + + - -

Immigration Law CasesImmigration Jurisdiction Im1 + + + + - + - - -Deporting Criminal Aliens Im2 + + + + + - - - -Detaining Criminal Aliens Im3 + + + + - + - - -Citizenship Im4 - - - + - + + + +

Speech and Press CasesLegal Aid for Poor SP1 + + + + - + - - -Privacy SP2 + + + + + + - - -Free Speech SP3 + - - - + + + + +Campaign Finance SP4 + + + + + - - - -Tobacco Ads SP5 - - - - + + + + +

Labor and Property Rights CasesLabor Rights LPR1 - - - - + + + + +Property Rights LPR2 - - - - + + + + +

The Supreme Court Justices andtheir ‘votes’ on a set of 26 “impor-tant decisions” made during the2000-2001 term, Doreian and Fu-jimoto (2002).The Justices (in the order inwhich they joined the SupremeCourt) are: Rehnquist (1972),Stevens (1975), O’Conner (1981),Scalia (1982), Kennedy (1988),Souter (1990), Ginsburg (1993)and Breyer (1994).



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. . . Supreme Court Voting / a (4,7) partition

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Open problems• GBM of valued networks

• GBM of multirelational networks

• GBM of temporal networks

• GBM of large networks



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GBM of valued networks

Can the clustering with relational constraint and blockmodeling problem begeneralized to a common problem?

Batagelj V., Ferligoj A.: Clustering relational data. Data Analysis (ed.: W.Gaul, O. Opitz, M. Schader), Springer, Berlin 2000, 3-15.


http://vlado.fmf.uni-lj.si/pub/preprint/HHBock.pdf


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General problem of clustering relational dataThe relationally constrained clustering problem with simple criterionfunction considers only the diagonal blocks that should be of one of thetypes Φi(R) . It also takes into account the dissimilarity matrix on units(derived from attribute data).

The blockmodeling problem deals only with relational data. The proposedoptimization approach essentially expresses the constraints with a penaltyfunction.

Both problems can be expressed as special cases of a clustering problemwith a general criterion function of the form

• G1s. P (C) =∑

(C1,C2)∈C×C q(C1, C2), or

• G1m. P (C) = max(C1,C2)∈C×C q(C1, C2)

where q is a block error satisfying

• G2. q(C1, C2) ≥ 0



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. . . clustering relational data

The set of feasible clusterings Φk(R) for this problem is determined by therelation R and additional requirements, such as:

• the blocks should be of selected types

• the model graph should be of specified form (prespecified)

• selected units should / should not be in the same cluster

• selected unit should / should not be in the selected cluster



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Approaches to the problem

There are different types of relational data (valued networks). In thefollowing we shall assume N = (V,R, a, b) where a : V → A assigns avalue to each unit/vertex and b : R → B assigns a value to each arc (link)of R. A and B are sets of values.

The function b determines a matrix B = [bij ]n×n, bij ∈ B ∪ {0} andbij = 0 if units i and j are not connected by an arc.

We can approach the problem of clustering relational data by indirect(transformation to standard data analysis problems) or direct (formulatingthe problem as an optimization problem and solving it) approach.



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Indirect approach

A scenario for the indirect approach is to transform attribute data a intodissimilarity matrix Da and network data b into dissimilarity matrix Db

and build criterion functions Pa and Pb based on them (they can be definedalso directly from a and b). Then we apply the multicriteria relationallyconstrained clustering methods on these functions.

We can also first combine Da and Db into a joint matrix Dab and applyrelationally constrained clustering methods on it.

In a special case, when Db is defined as some corrected dissimilarity(see Batagelj, Ferligoj, Doreian, 1992) between descriptions b(x) =[B(x),BT (x)], the relational data are built into Db and we can apply onthe combined matrix Dab all standard methods for analysis of dissimilaritymatrices.



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Direct approach

Again there are different possibilities:

1. Structural approach: used in program MODEL (Batagelj, 1996):Important is the structure (relation). Determine the best clustering Cand the corresponding model. On the basis of a, b and the obtainedmodel compute values of model connections.

2. Multicriteria approach: construct two criterion functions: one based onvalues, the second based on structure. Solve the obtained multicriteriaproblem (Ferligoj, Batagelj, 1992).

3. Implicit approach: the types of connections are built into the criterionfunction combined with values.

Only the last approach needs some further explanations.



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Implicit approach

Let ≈ be an equivalence over set of units V , and T given types. Weconstruct on blocks deviation functions δ(C1, C2;T ), T ∈ T such that ≈ iscompatible with T over the network N iff

∀u, v ∈ V ∃δ(., .;T ), T ∈ T : δ([u], [v];T ) = 0

Applying also an adequate normalization of δs we can construct a criterionfunction

P (C) =∑

C1,C2∈C

minT∈T

δ(C1, C2;T )

Evidently, P (C) = 0 ⇔ C is compatible with T — all blocks of C arecompatible with T .



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Some examples

Assume that a and b are transformed into a matrix A = [auv]V×V , auv ≥ 0.Then

δ(X, Y ; nul) =

∑x∈X,y∈Y axy

|X| · |Y | ·max{axy : x ∈ X, y ∈ Y }

δ(X, Y ; rdo) = 1−maxx∈X

∑y∈Y axy

|Y | ·max{axy : y ∈ Y }

δ(X, Y ; cre) = 1−∑

y∈Y maxx∈X axy

|Y | ·max{axy : x ∈ X, y ∈ Y }

If max in the denominator equals to 0 then also the fraction has value 0.

Some ideas how to approach the GBM of valued networks will be presentedby Ales Ziberna.



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GBM of multirelational networksMultiple networks are networks with more than one relation defined on thesame set of vertices.

Assume that we have relations R1, R2, . . . , Rs and corresponding criterionfunctions P1, P2, . . . , Ps compatible respectively with T1, T2, . . . , Ts. Thenwe get the following multicriteria optimization problem: determine C? ∈ Φthat ’minimizes’

(Φ, P1, P2, . . . , Ps)

This problem can be approched by (Ferligoj, Batagelj, 1992)

• multicriteria optimization. The solutions are Pareto points.

• transformation to single criterion optimization

– by combining criterion functions:∑

αiPi or max αiPi, whereαi ≥ 0 and

∑αi = 1;

– by combining relations.



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GBM of temporal networks

If the clustering C is the same for all time points we can treat the GBMof temporal networks problem as a GBM of multirelational networks forrelations determined by the time slices.

In general, however, also the clustering C can change through time –the solution is a sequence of clusterings (C1,C2, . . . ,Cs), Ci ∈ Φi (orCi ∈ Φi(Ci−1)).



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The Sampson DataThe affect data from Sampson (1968) provide a useful source for examiningstructural balance through time. (See Doreian and Mrvar, 1996, for a morecomplete description and use of these data.) There are three subgroups:One group was the ‘Young Turks’ made up of Gregory, John Bosco, Mark,Winfrid, Hugh, Boniface and Albert.

A second subgroup was labeled the ‘Loyal Opposition’ and comprised Peter,Bonaventure, Berthold, Victor, Ambrose, Romuald, Louis and Amand.

The remaining three individuals - Basil, Simplicius and Elias - were labelledas the ‘Outcasts’.

We consider the partitions reported by Doreian and Mrvar (1996).

Sampson reported data for these actors for 3 periods in time. Each actorwas asked to name the three other actors he liked the most and the three hedisliked the most. The ties {3, 2, 1} are the labels for the positive ties while{-3, -2, -1} denote the negative ties.



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k T2 T3 T4

1 48.5 48.0 47.0

2 21.5 16.0 12.5

3 17.5 11.0 10.5

4 19.0 13.5 12.5

5 20.5 16.0 15.0

Table reports the values of the criterion function for partitions having 1 to5 clusters. Partitions consistent with structural balance (into two plus-sets)show a consistent drop through time of the criterion function. The same istrue for generalized of balance (k > 2). The lowest values of the criterionfunction are those for a partition into 3 plus-sets. table shows strong supportfor the fundamental structural balance hypothesis. This group evolvedthrough time towards a balanced form.

Partition of Sampson Monks at T2, T3, and T4



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T21 2 7 12 14 15 16 3 13 17 18 4 5 6 8 9 10 11

John Bosco 1 · · -1 · 1 · · 2 · · · · 3 -2 · · -3 ·Gregory 2 3 · 2 · 1 · · · -3 -2 · · · · · · -1 ·Mark 7 · 2 · · · · 3 · · · · -3 -1 -2 1 · · ·Winfrid 12 3 2 · · 1 · · -1 · · · -3 · · · · · -2Hugh 14 3 · · 2 · 2 · · -3 -1 · · · · -2 · · 1Boniface 15 3 2 · · 1 · · -2 -3 -1 -1 · · · · · · ·Albert 16 1 2 3 · · · · · -1 -3 -2 · · · · · · ·Basil 3 2 3 · · · · · · · 1 · -1 · · -3 -2 · ·Amand 13 · -3 1 -1 · · · · · · 3 · 2 -2 · · · ·Elias 17 · · · · · · · 3 2 · 1 -3 -2 · · · · -1Simplicius 18 2 3 1 · · · -1 · · · · -3 · -2 · · · ·Peter 4 · · -3 · · · · -2 · · -1 · 3 1 · · 2 ·Bonaventure 5 · · · · · · · · 1 · · 3 · · · · · 2Berthold 6 1 · -3 -2 · · · · · · · 3 · · -1 2 · ·Victor 8 3 2 · · -2 · · -3 · -1 · · · · · 1 · ·Ambrose 9 · · · · · · 1 -3 · -2 -1 · 2 · 3 · · ·Romuald 10 · · · · 2 · · · · · · 3 · · 1 · · ·Louis 11 · · · · 2 · · -1 -3 -2 · · 3 · 1 · · ·



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T31 2 7 12 14 15 16 3 13 17 18 4 5 6 8 9 10 11

John Bosco 1 · 2 · · · · · · · · · -2 1 · 3 · -3 -1Gregory 2 3 · 1 2 · 1 · · -1 · · -3 · · -2 · · ·Mark 7 1 2 · · · · 3 · · · · -3 · -2 -1 · · ·Winfrid 12 3 1 · · 2 · · -3 · -1 · · · -2 · · · ·Hugh 14 3 1 · 1 · 2 · · · -1 · -3 · · · · -2 ·Boniface 15 2 3 · · 1 · · -2 -3 -1 · · 1 · · · -1 ·Albert 16 · 2 3 1 · · · -1 · -2 · -3 · · · · · ·Basil 3 3 -1 · · · · · · · 1 2 · · -2 -3 · · ·Amand 13 · -3 1 -1 · · · · · · 3 · 2 -2 · · · ·Elias 17 · 1 · · · · · 2 · · 3 -2 · -1 · · · -3Simplicius 18 · 1 · · · · · · 2 3 · -3 · -2 · · · -1Peter 4 -2 -3 · · · · · · · · · · 3 2 · · · 1Bonaventure 5 2 · · · · · · · · · · 3 · · · · · 1Berthold 6 1 · · -2 · · · -3 · · · 3 · · -1 2 · ·Victor 8 -2 -3 · · · · · -1 · · · 3 · 1 · · 2 ·Ambrose 9 · · · 2 · · · -3 · -2 -1 · 1 · 3 · · ·Romuald 10 · · · · · · · · 2 · · 3 1 · · · · ·Louis 11 -1 -3 · · · · 1 -2 · · · 2 3 · · · · ·



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T41 2 7 12 14 15 16 3 13 17 18 4 5 6 8 9 10 11

John Bosco 1 · -2 -3 1 2 · · 3 · · · · · · · · -1 ·Gregory 2 3 · 1 2 · · · · -1 · · -3 · · -2 · · ·Mark 7 · 3 · 1 · · 2 · · · · -3 · -2 -1 · · ·Winfrid 12 3 2 1 · · · · · · · · · · · · · · ·Hugh 14 3 · · 1 · 2 · · · -1 · -3 · · -2 · · ·Boniface 15 · 3 1 2 · · · -2 -3 · · -1 · · · · · ·Albert 16 · 3 2 · · 1 · -1 · -2 · -3 · · · · · ·Basil 3 3 -2 · · · · -1 · 2 1 2 -3 · -2 · · · ·Amand 13 · -3 1 -1 · · · · · · 3 · 2 -2 · · · ·Elias 17 · 1 · · · · · 2 · · 3 -1 · -3 -2 · · ·Simplicius 18 · 1 · · · · · 2 · 3 · -1 · · -3 · -2 ·Peter 4 -2 -3 · · -1 · · · · · · · 3 1 · · · 2Bonaventure 5 · · · · · · · · · · · 3 · · · 1 · 2Berthold 6 · -1 -2 · · · · -3 · -2 · 3 1 · · 2 · ·Victor 8 · -3 · · -1 · · -2 · · · 3 · 2 · 1 · ·Ambrose 9 · · · 2 · · · -3 · -2 -1 · 1 · 3 · · ·Romuald 10 · · · · · · · · 2 · · 3 1 · · 1 · ·Louis 11 -1 -3 · · 1 · · -2 · · · · 2 · 3 · · ·



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GBM of large networks

Large networks are usually sparse m << n2.

In the case of given partition C (for example partition of contries to conti-nents, patents to categories, . . . ) it is easy to determine the correspondingGBM that minimizes the criterion function.

A special class of GBM problems are symmetric-acyclic decompositions(Doreian, Batagelj, Ferligoj, 2000) for which also an algorithm for largenetworks was developed.



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. . . GBM of large networks

For some GBM problems on large networks the clustering methods can beused:

• if we have to compute the dissimilarities between attribute data forvertices we may consider methods that require only dissimilaritiesbetween the linked vertices;

• if we base the clustering on the descriptions of vertices using selectedstructural properties we can consider some variant of leader’s method.


advances in generalized blockmodeling pat doreianvlado.fmf.uni-lj.si/pub...

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